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Hodge Theory and Moduli Contemporary Mathematics Volume 766, 2021 https://doi.org/10.1090/conm/766/15381 Hodge theory and moduli Phillip Griffiths Outline I.A Introductory comments I.B Classification problem; first case of relation between moduli and Hodge theory II.A Moduli II.B Hodge theory III.A Generalities on Hodge theory and moduli III.B I-surfaces and their period mappings Appendix; Application of Hodge theory to I-surfaces with a degree 2 el- liptic singularity I.A. Introductory comments. Algebraic geometry is frequently seen as a very interesting and beautiful subject, but one that is also very difficult to get into. This is partly due to its breadth, as traditionally algebra (commutative and homological), topology, analysis, differential geometry, Lie theory — and more recently combinatorics, logic, categorical and derived algebraic techniques, . — are used to study it. I have tried to make these notes accessible to a general audience by reviewing some of the most basic concepts, illustrating the material with elementary examples and informal geometric and heuristic arguments, and with occasional side comments for experts in the subject. Although algebra, both commutative and homological, are the central tools in algebraic geometry, their use in many current areas of research (birational geometry and the minimal model program) is frequently quite technical and will not be extensively discussed in these notes. On the other hand, partly because Hodge theory involves analysis, Lie theory and differential geometry as well as a wide variety of homological methods, it is perhaps less prevalent in the more algebraic works in the field. One objective of these talks is to illustrate how, in partnership with the more algebraic and homological methods, Hodge theory may be used to study interesting and important geometric questions. In the appendix we have sketched how this may be carried out in a particular example. Notes based on the lecture presented at the conference “Geometry at the frontier” held at Pucon, Chile during November 2018. Parts of this paper represents joint work in progress with Mark Green, Radu Laza and Colleen Robles. c 2021 American Mathematical Society 163 This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 164 PHILLIP GRIFFITHS In summary the theme of these notes is to discuss and illustrate how complex analysis, differential geometry and Lie theory may be combined to study a basic problem in algebraic geometry. Although the main techniques in much of contem- porary algebraic geometry are algebraic, some of the most interesting questions in the subject require non-algebraic methods for their study and this is what we hope to illustrate.1 As a warm up and to establish some notations we recall that an affine algebraic variety is given by the solution space over the complex numbers to polynomial equations (∗) fi(x1,...,xn)=0 i =1,...,m. The “elementary” examples are ¨¨ linear spaces ax + by + c =0 ¨ conics ax2 +2bxy + cy2 + ex + fy + g =0 n n quadrics Q(x)= aijxixj + bixi + c =0,aij,aji i,j=1 i=1 The first non-elementary examples are cubics y2 =4x3 + ax + b The first two elementary examples and the non-elementary example are algebraic curves. One of course considers higher dimensional varieties; surfaces, threefolds,. We will be primarily concerned with curves and surfaces. The above algebraic curves are all affine algebraic varieties in C2. In general one adds to an affine variety the asymptotes or “points at infinity” to obtain the n n n−1 n n+1 projective space P = C ∪ P∞ .Equivalently,P is the quotient of C \{0} ∗ n by the scaling action zi → λzi where λ ∈ C . Geometrically P is the set of lines through the origin in Cn+1. The picture of the projective plane P2 is line at infinity 1This paper is written in an informal style, usually without precise definitions and statements of results. At the end there is a fairly extensive set of references where the more standard presen- tation of the material to algebraic geometers may be found. In particular we suggest [CMSP17] as a general reference relating Hodge theory and algebraic geometry and where several examples relevant to these notes are discussed in detail. This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. HODGE THEORY AND MODULI 165 and the picture in P2 of the hyperbola in C2 given by y x2 − y2 =1 x is something like The equations of the completion in Pn of an affine variety given in Cn by (∗) are obtained by homogenizing: set xi = zi/z0 and clear denominators to obtain fi(z0,z1,...,zn)=0 di where fi(λz0,λz1,...,λzn)=λ fi(z0,zi,...,zn), di =degfi. Then the hyperbola above becomes 2 − 2 2 z1 z2 = z0 . Later on we will consider varieties in weighted projective spaces P(a0,a1,...,an) ∗ ai n+1 ∗ where λ ∈ C acts on zi by λ(zi)=λ zi, and in the quotient C \{0}/C the algebraic varieties are defined by weighted homogeneous polynomials. Here the weights ai are positive integers and gcd(a1,...,aN ) = 1. The following will be used in our discussion of the I-surfaces: Example: P(1, 1, 2) is embedded in P3 by → 2 2 [x0,x1,y] [x0,x0x1,x1,y]. The image is 2 z0z2 = z1 , which is Two algebraic varieties are considered to be equivalent if there is a “change of variables” that transforms one into the other. Thus if the discriminant b2 − ac =0 all conics are equivalent to the circle 2 2 2 z1 + z2 = z0 . Initially changes of variables were linear transformations (including projections); later on rational changes of variables became allowed. Historically algebraic varieties arose from two sources: projective geometry (lines and linear spaces, conics and higher dimensional quadrics), and from complex analysis. In complex analysis the issue was to understand the integrals r(x, y(x))dx, f(x, y(x)) = 0 of algebraic functions, and the “functions” defined by inverting the integrals x(w) w = r(x, y(x))dx. Here f(x, y) is a polynomial and r(x, y) is a rational function; y(x) is a multi- valued “algebraic function” defined by f(x, y(x)) = 0. The integral depends on This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. 166 PHILLIP GRIFFITHS choosing a path γ of integration in the complex plane and a branch y(x)off(x, y(x)) = 0 along γ.2 Thus sin w dx sin w dx w = √ = 1 − x2 y(x) where x2 + y2 =1and dx2 + dy2 = dx/y gives the parametrization w → (sin w, cos w) of the circle by arclength in terms of “elementary” functions (trigono- metric functions and logarithms). However parametrizing the ellipse by arclength led to integrals such as p(w) dx w = √ 4x3 + ax + b which gave non-elementary functions and led Euler, Legendre, Abel, Gauss, Jacobi, Riemann,. to the beginnings of the rich and deep interplay between analysis and algebraic geometry. This evolved into modern Hodge theory, and it is this interface between analysis and algebraic geometry that is a main theme of these talks. Part of the richness of the subject of algebraic geometry are the multiple per- spectives that may be used in its study: • geometric; • algebraic — e.g., as we will briefly discuss, in birational geometry the algebraic classification of certain classes of singularities of algebraic varieties plays a central role;3 • analytic; we have mentioned complex analysis and the integrals of algebraic functions; • topological; one always has in mind the first picture that is encountered. Example (running example for illustrative purposes): The compact 1- dimensional complex manifold associated to the algebraic curve 2 X = {y =(x − a1)(x − a2) ···(x − a2g+1)} where the ai are distinct is a compact Riemann surface of genus g which can be studied from all of these perspectives. 2A more correct notation would be w = (x(w),y(w)) r(x, y(x))dx where (x(w),y(w)) is the point on the curve f(x, y)=0inC2 that is given by the integral along the path γ in the curve. To simplify the notation the second coordinate y(w) will be implicit in what follows. 3Cf. [Kol13b] in which the extent and complexity of this story are on full display. This is a free offprint provided to the author by the publisher. Copyright restrictions may apply. HODGE THEORY AND MODULI 167 For g = 1 as in the cubic above, in addition to the topological one the pictures are algebraic + ∞ a1 a2 a3 analytic - ∞ where a1,a2,a3 are the roots of the cubic equation in the RHS of the above equation. The + and − represent the two possible values of y(x) with the un- derstanding that analytic construction across a slit changes the sign of y(x)= (x − a1)(x − a2) ···(x − a2g+1). The integral w dx y is determined up to the periods dx dx π1 = ,π2 = ; δ y γ y one may show that π1 =0andIm( π2/π1) > 0. Incidentally for the hyperbola y2 = x2 − 1 the picture is + - ; dx there is only one cycle δ. In this case there is only one period δ y . For another analytic perspective, as will be further discussed below the function p(w) given by inverting the elliptic integral is a doubly periodic entire function 4 (doubly due to the periods δ dx/y and γ dx/y) that leads to the parametrization of the cubic curve C /X ∈ ∈ w /(p(w),p(w)) 4For the hyperbola above there is only one period and inverting the integral gives singly periodic trigonometric functions.
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